## The Annals of Statistics

### A complementary set theory for quaternary code designs

#### Abstract

Quaternary code (QC) designs form an attractive class of nonregular factorial fractions. We develop a complementary set theory for characterizing optimal QC designs that are highly fractionated in the sense of accommodating a large number of factors. This is in contrast to existing theoretical results which work only for a relatively small number of factors. While the use of imaginary numbers to represent the Gray map associated with QC designs facilitates the derivation, establishing a link with foldovers of regular fractions helps in presenting our results in a neat form.

#### Article information

Source
Ann. Statist., Volume 41, Number 6 (2013), 2768-2785.

Dates
First available in Project Euclid: 17 December 2013

https://projecteuclid.org/euclid.aos/1387313389

Digital Object Identifier
doi:10.1214/13-AOS1160

Mathematical Reviews number (MathSciNet)
MR3161447

Zentralblatt MATH identifier
1292.62119

Subjects
Primary: 62K15: Factorial designs

#### Citation

Mukerjee, Rahul; Tang, Boxin. A complementary set theory for quaternary code designs. Ann. Statist. 41 (2013), no. 6, 2768--2785. doi:10.1214/13-AOS1160. https://projecteuclid.org/euclid.aos/1387313389

#### References

• [1] Box, G. and Tyssedal, J. (1996). Projective properties of certain orthogonal arrays. Biometrika 83 950–955.
• [2] Chen, H. and Hedayat, A. S. (1996). $2^{n-l}$ designs with weak minimum aberration. Ann. Statist. 24 2536–2548.
• [3] Cheng, C.-S. and Mukerjee, R. (1998). Regular fractional factorial designs with minimum aberration and maximum estimation capacity. Ann. Statist. 26 2289–2300.
• [4] Cheng, S.-W., Li, W. and Ye, K. Q. (2004). Blocked nonregular two-level factorial designs. Technometrics 46 269–279.
• [5] Deng, L.-Y. and Tang, B. (1999). Generalized resolution and minimum aberration criteria for Plackett–Burman and other nonregular factorial designs. Statist. Sinica 9 1071–1082.
• [6] Mukerjee, R. and Wu, C. F. J. (2006). A Modern Theory of Factorial Designs. Springer, New York.
• [7] Phoa, F. K. H. (2012). A code arithmetic approach for quaternary code designs and its application to $(1/64)$th-fractions. Ann. Statist. 40 3161–3175.
• [8] Phoa, F. K. H., Mukerjee, R. and Xu, H. (2012). One-eighth- and one-sixteenth-fraction quaternary code designs with high resolution. J. Statist. Plann. Inference 142 1073–1080.
• [9] Phoa, F. K. H. and Xu, H. (2009). Quarter-fraction factorial designs constructed via quaternary codes. Ann. Statist. 37 2561–2581.
• [10] Tang, B. and Deng, L.-Y. (1999). Minimum $G_{2}$-aberration for nonregular fractional factorial designs. Ann. Statist. 27 1914–1926.
• [11] Tang, B. and Wu, C. F. J. (1996). Characterization of minimum aberration $2^{n-k}$ designs in terms of their complementary designs. Ann. Statist. 24 2549–2559.
• [12] Wu, C. F. J. and Hamada, M. S. (2009). Experiments: Planning, Analysis, and Optimization, 2nd ed. Wiley, Hoboken, NJ.
• [13] Xu, H. (2003). Minimum moment aberration for nonregular designs and supersaturated designs. Statist. Sinica 13 691–708.
• [14] Xu, H. (2009). Algorithmic construction of efficient fractional factorial designs with large run sizes. Technometrics 51 262–277.
• [15] Xu, H., Phoa, F. K. H. and Wong, W. K. (2009). Recent developments in nonregular fractional factorial designs. Stat. Surv. 3 18–46.
• [16] Xu, H. and Wong, A. (2007). Two-level nonregular designs from quaternary linear codes. Statist. Sinica 17 1191–1213.
• [17] Xu, H. and Wu, C. F. J. (2001). Generalized minimum aberration for asymmetrical fractional factorial designs. Ann. Statist. 29 1066–1077.
• [18] Zhang, R., Phoa, F. K. H., Mukerjee, R. and Xu, H. (2011). A trigonometric approach to quaternary code designs with application to one-eighth and one-sixteenth fractions. Ann. Statist. 39 931–955.